ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brcogw Unicode version

Theorem brcogw 4768
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
brcogw  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )

Proof of Theorem brcogw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 989 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A  e.  V )
2 simpl2 990 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  B  e.  W )
3 breq2 3981 . . . . . 6  |-  ( x  =  X  ->  ( A D x  <->  A D X ) )
4 breq1 3980 . . . . . 6  |-  ( x  =  X  ->  (
x C B  <->  X C B ) )
53, 4anbi12d 465 . . . . 5  |-  ( x  =  X  ->  (
( A D x  /\  x C B )  <->  ( A D X  /\  X C B ) ) )
65spcegv 2810 . . . 4  |-  ( X  e.  Z  ->  (
( A D X  /\  X C B )  ->  E. x
( A D x  /\  x C B ) ) )
76imp 123 . . 3  |-  ( ( X  e.  Z  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
873ad2antl3 1150 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
9 brcog 4766 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
109biimpar 295 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  E. x
( A D x  /\  x C B ) )  ->  A
( C  o.  D
) B )
111, 2, 8, 10syl21anc 1226 1  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 967    = wceq 1342   E.wex 1479    e. wcel 2135   class class class wbr 3977    o. ccom 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-co 4608
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator